scholarly journals Stable Equivalence over Symmetric Functions

10.37236/1880 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Arthur L. B. Yang
Keyword(s):  
Symmetric Functions ◽  
Affirmative Answer ◽  
Schur Function ◽  
Stable Equivalence ◽  
Skew Schur Function

By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli-type matrices for a given skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the transpose of the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This leads to an affirmative answer to a question proposed by Kuperberg.

scholarly journals A Hopf Algebraic Approach to Schur Function Identities

10.37236/5741 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Karen Yeats
Keyword(s):  
Hopf Algebra ◽  
Symmetric Functions ◽  
Algebraic Approach ◽  
Schur Functions ◽  
Schur Function ◽  
Skew Schur Function ◽  
Skew Schur Functions

Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.

scholarly journals Lattice Path Proofs of Extended Bressoud-Wei and Koike Skew Schur Function Identities

10.37236/534 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
A. M. Hamel ◽  
R. C. King
Keyword(s):  
Lattice Path ◽  
Schur Function ◽  
Skew Schur Function

A recent paper of the present authors provides extensions to two classical determinantal results of Bressoud and Wei, and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs.

scholarly journals Crystals and Schur $P$-Positive Expansions

10.37236/7557 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Seung-Il Choi ◽  
Jae-Hoon Kwon
Keyword(s):  
Crystal Structures ◽  
Schur Function ◽  
Skew Schur Function

We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mathfrak{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.

On the product of a Schur function and a skew Schur function

1993 ◽  
Vol 34 (2) ◽  
pp. 305-315
Author(s):  
Toshihiro Watanabe
Keyword(s):  
Schur Function ◽  
Skew Schur Function

scholarly journals Extended Bressoud–Wei and Koike skew Schur function identities

2011 ◽  
Vol 118 (2) ◽  
pp. 545-557
Author(s):  
A.M. Hamel ◽  
R.C. King
Keyword(s):  
Schur Function ◽  
Skew Schur Function

scholarly journals Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets

10.37236/1264 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Partially Ordered Sets ◽  
Schur Function ◽  
Ordered Sets ◽  
Graded Poset ◽  
Kostka Polynomials ◽  
Relative Version ◽  
Definition Of ◽  
Formal Power

For every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ we associate a certain formal power series $F_P(x) = F_P(x_1,x_2,\dots)$ which encodes the flag $f$-vector (or flag $h$-vector) of $P$. A relative version $F_{P/\Gamma}$ is also defined, where $\Gamma$ is a subcomplex of the order complex of $P$. We are interested in the situation where $F_P$ or $F_{P/\Gamma}$ is a symmetric function of $x_1,x_2,\dots$. When $F_P$ or $F_{P/\Gamma}$ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called $q$-primary lattices the Schur function coefficients are just values of Kostka polynomials at the prime power $q$, thus giving in effect a simple new definition of Kostka polynomials in terms of symmetric functions. We extend the theory of lexicographically shellable posets to the relative case in order to show that some examples $(P,\Gamma)$ are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.

scholarly journals Two Remarks on Skew Tableaux

10.37236/2012 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Generating Functions ◽  
Schur Function ◽  
Young Tableaux ◽  
Skew Schur Function

This paper contains two results on the number $f^{\sigma/\tau}$ of standard skew Young tableaux of shape $\sigma/\tau$. The first concerns generating functions for certain classes of "periodic" shapes related to work of Gessel-Viennot and Baryshnikov-Romik. The second result gives an evaluation of the skew Schur function $s_{\lambda/\mu}(x)$ at $x=(1,1/2^{2k},1/3^{2k}, \dots)$ for $k=1,2,3$ in terms of $f^{\sigma/\tau}$ for a certain skew shape $\sigma/\tau$ depending on $\lambda/\mu$.

scholarly journals Grothendieck Bialgebras, Partition Lattices, and Symmetric Functions in Noncommutative Variables

10.37236/1101 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
N. Bergeron ◽  
C. Hohlweg ◽  
M. Rosas ◽  
M. Zabrocki
Keyword(s):  
Symmetric Functions ◽  
Schur Function ◽  
Partition Lattice ◽  
Partition Lattices

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.

scholarly journals A Macdonald Vertex Operator and Standard Tableaux Statistics

10.37236/1383 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mike Zabrocki
Keyword(s):  
Vertex Operator ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Function ◽  
Inductive Proof ◽  
Integer Coefficients ◽  
The Family ◽  
The Relationship ◽  
Two Parameter

The two parameter family of coefficients $K_{\lambda \mu}(q,t)$ introduced by Macdonald are conjectured to $(q,t)$ count the standard tableaux of shape $\lambda $. If this conjecture is correct, then there exist statistics $a_\mu(T)$ and $b_\mu(T)$ such that the family of symmetric functions $H_\mu[X;q,t] = \sum_\lambda K_{\lambda \mu}(q,t) s_\lambda [X]$ are generating functions for the standard tableaux of size $|\mu|$ in the sense that $H_\mu[X;q,t] = \sum_{T} q^{a_\mu(T)} t^{b_\mu(T)} s_{\lambda (T)}[X]$ where the sum is over standard tableau of of size $|\mu|$. We give a formula for a symmetric function operator $H_2^{qt}$ with the property that $H_2^{qt} H_{(2^a1^b)}[X;q,t]= H_{(2^{a+1}1^b)}[X;q,t]$. This operator has a combinatorial action on the Schur function basis. We use this Schur function action to show by induction that $H_{(2^a1^b)}[X;q,t]$ is the generating function for standard tableaux of size $2a+b$ (and hence that $K_{\lambda (2^a1^b)}(q,t)$ is a polynomial with non-negative integer coefficients). The inductive proof gives an algorithm for 'building' the standard tableaux of size $n+2$ from the standard tableaux of size $n$ and divides the standard tableaux into classes that are generalizations of the catabolism type. We show that reversing this construction gives the statistics $a_\mu(T)$ and $b_\mu(T)$ when $\mu$ is of the form $(2^a1^b)$ and that these statistics prove conjectures about the relationship between adjacent rows of the $(q,t)$-Kostka matrix that were suggested by Lynne Butler.

scholarly journals The Murnaghan―Nakayama rule for k-Schur functions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Anne Schilling ◽  
Mike Zabrocki
Keyword(s):  
Explicit Formula ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Schur Function ◽  
Noncommutative Symmetric Functions ◽  
Power Sum ◽  
International Audience

International audience We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene. Nous prouvons une règle de Murnaghan-Nakayama pour les fonctions de k-Schur de Lapointe et Morse, c'est-à-dire que nous donnons une formule explicite pour le développement du produit d'une fonction symétrique "somme de puissances'' et d'une fonction de k-Schur en termes de fonctions k-Schur. Ceci est prouvé en utilisant les fonctions non commutatives k-Schur en termes d'algèbre nilCoxeter introduite par Lam et l'analogue affine des fonctions symétriques non commutatives de Fomin et Greene.

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